The Studierendenwerk Hamburg is responsible for processing your BAföG application. The earlier you apply, the likelier you are to receive funding from the beginning of the semester. BAföG is not, under any circumstances, paid retroactively.
As a rule, half of the German monthly BAföG student loan is in effect a grant. The other half is an interest-free loan that must be repaid when you have completed your course of study. In addition to BAföG you can also apply for an educational loan or a bank loan on which interest is payable.
The necessary document (§ 9 BAföG-Certificate) in available by using the online services for students.
For everything you need to know, try the following links:
Studierendenwerk Hamburg (Application!)